Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913803720982528 |
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| author | Kline, Josh Li, Feng Shanmugalingam, Nageswari |
| author_facet | Kline, Josh Li, Feng Shanmugalingam, Nageswari |
| contents | In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,ν)$ equipped with a doubling measure $ν$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $π$ which is in a $σ$-codimensional relationship to $ν$ for some $σ>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces Kline, Josh Li, Feng Shanmugalingam, Nageswari Analysis of PDEs Metric Geometry 46E36, 31E05, 35R11, 60G22 In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,ν)$ equipped with a doubling measure $ν$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $π$ which is in a $σ$-codimensional relationship to $ν$ for some $σ>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous. |
| title | Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces |
| topic | Analysis of PDEs Metric Geometry 46E36, 31E05, 35R11, 60G22 |
| url | https://arxiv.org/abs/2408.02624 |